BSEH Class 12 Maths key terms

A relation R on a set A where every element is related to itself, meaning (a, a) belongs to R for every a in A. A set with no elements satisfies this automatically.

A relation that is reflexive, symmetric and transitive at the same time. It splits the underlying set into disjoint equivalence classes whose members are all related to one another.

A relation R on a set A where whenever (a, b) belongs to R, then (b, a) also belongs to R. The order of the pair can be reversed without leaving the relation.

A relation R on a set A where (a, b) and (b, c) in R always force (a, c) into R. Chaining two related pairs never produces a pair outside the relation.

A function that is both one-one (injective) and onto (surjective), so each element of the codomain is matched with exactly one element of the domain. Only bijections possess an inverse function.

A function f where distinct inputs always give distinct outputs, that is f(x1) = f(x2) implies x1 = x2. Also called injective; its graph is never crossed twice by a horizontal line.

The function f inverse that undoes f, defined only when f is bijective, by swapping domain and codomain. Its graph is the mirror image of the graph of f in the line y = x.

The function formed by applying one function to the output of another, written (g of f)(x) = g(f(x)). It exists only when the range of f lies inside the domain of g.

The restricted range chosen for an inverse trigonometric function so that it becomes one-one and its inverse is a genuine function. For sine it is minus pi by 2 to pi by 2 inclusive.

The set of inputs accepted by inverse sine, namely the closed interval from minus 1 to 1. Values outside this interval have no real angle whose sine they equal.

The principal output interval of inverse cosine, the closed interval from 0 to pi. Choosing this branch makes cosine one-one and keeps the inverse function single-valued.

The addition formula arctan x plus arctan y equals arctan of (x plus y) divided by (1 minus xy), valid when xy is less than 1. A sign adjustment is needed when xy exceeds 1.

A square matrix equal to its own transpose, so the entry in row i column j equals the entry in row j column i. Its diagonal entries are unrestricted.

A square matrix whose transpose is its negative, forcing every diagonal entry to be zero. Written A transpose equals minus A; it has odd order determinant zero.

The matrix obtained by turning rows into columns and columns into rows, written A transpose. Its order is reversed, and transposing twice returns the original matrix.

A square matrix whose determinant equals zero, so it has no multiplicative inverse. Its rows or columns are linearly dependent and it collapses area or volume.

A square matrix with non-zero determinant that possesses an inverse satisfying A times A inverse equals the identity. Its inverse is adjoint of A divided by determinant of A.

The transpose of the cofactor matrix of a square matrix, written adj A. Multiplying a matrix by its adjoint gives the identity scaled by the determinant.

The signed minor of an entry, obtained by deleting its row and column and multiplying the remaining determinant by plus or minus one according to position. Cofactors build the adjoint and the determinant expansion.

The determinant left after removing the row and column containing a chosen entry of a square matrix. Cofactors attach a checkerboard sign to these minors.

A single number computed from a square matrix that measures how the associated transformation scales area or volume. It is zero exactly when the matrix is singular.

A function is continuous at a point when its limit there exists, the function is defined there, and the two values agree. All three conditions must hold simultaneously.

A function is differentiable at a point when its derivative exists there, meaning the difference quotient has a single finite limit. Differentiability implies continuity, but continuity does not imply differentiability.

The derivative measures how fast a quantity changes with respect to another, equal to the slope of the tangent line. For position against time it gives instantaneous velocity.

An interior point of the domain where the derivative is zero or fails to exist. Local maxima and minima of a differentiable function can occur only at such points.

A method classifying critical points by the sign change of the derivative: positive to negative gives a local maximum, negative to positive gives a local minimum. No sign change gives neither.

A method classifying a critical point using the second derivative: negative means local maximum, positive means local minimum. A zero value leaves the test inconclusive.

The largest value a function attains over its whole stated domain. On a closed interval it occurs at a critical point or at an endpoint.

A technique reversing the product rule, written integral of u dv equals uv minus integral of v du. Choosing u as the function that simplifies on differentiation makes it effective.

A technique where a new variable replaces an inner expression, rewriting the integral into a simpler standard form. The differential must be converted consistently with the substitution.

Splitting a rational integrand into a sum of simpler fractions with linear or quadratic denominators. Each piece then matches a standard logarithmic or inverse tangent integral.

The limit of a Riemann sum over an interval, interpreted as signed area between the curve and the axis. It is evaluated using a primitive and the two endpoint values.

The region bounded by a curve, the x-axis and two vertical lines, found by integrating the function between the limits. Portions below the axis contribute negatively unless absolute value is taken.

The region enclosed by two graphs, computed by integrating the upper function minus the lower function between their intersection points. Intersections must be found before setting limits.

The order is the highest derivative appearing in the equation. It is unrelated to the power of that derivative, which defines the degree instead.

The power of the highest order derivative once the equation is written as a polynomial in its derivatives. Undefined when derivatives appear inside transcendental functions.

A multiplier that turns a first-order linear differential equation into an exact one, equal to e raised to the integral of the coefficient of y. Multiplying both sides makes the left side a derivative of a product.

The cosines of the angles a vector or line makes with the positive coordinate axes. Their squares sum to one, and they fix direction but not magnitude.

The scalar obtained by multiplying corresponding components and adding, also equal to the product of magnitudes times the cosine of the angle between them. It is zero for perpendicular vectors.

A vector perpendicular to two given vectors, with magnitude equal to their magnitudes times the sine of the angle between them. It vanishes when the vectors are parallel.

The dot product of one vector with the cross product of two others, giving the signed volume of the parallelepiped they span. It is zero exactly when the three vectors are coplanar.

The distance between two non-intersecting, non-parallel lines, found by projecting the vector joining a point on each line onto their common normal. It uses the scalar triple product over the cross product magnitude.

The set of points satisfying every constraint of a linear programming problem, usually a polygon in the plane. The optimum of a linear objective over it occurs at a corner point.

A procedure evaluating the objective function at every vertex of the feasible region and picking the best value. The corner point theorem guarantees an optimum lies at a vertex.

The linear expression whose value is to be maximised or minimised in a linear programming problem, subject to the constraints. Its coefficients come from per-unit profit or cost.

The probability of an event given that another event has already occurred, written P of A given B and equal to P of A and B divided by P of B. It requires P of B to be non-zero.

Two events where the occurrence of one does not change the probability of the other, so P of A and B equals P of A times P of B. This differs from mutually exclusive events.

A rule revising probabilities in light of new evidence, expressing P of A given B through the reverse conditional and prior probabilities. It is used in problems on false positives and disease testing.

The probability distribution of the number of successes in a fixed number of independent trials with constant success probability. Its mean is n times p and variance is n times p times q.